To complement Alexandre's answer, there is an application of SCV in physics, in quantum field theory. One can prove that certain quantities ($n$-point functions $G(x_1,\ldots,x_n)$) appearing in (free) quantum field models are analytic in their $n$ $R^4$ arguments (while their Fourier transforms are essentially hyperfunctions). The analytic continuation into $n$ $\mathbb{C}^4$ arguments of the $n$-point functions and their Fourier transforms then allows one to prove some physically important results. For example, the proof of the spin-statistics theorem (fermions have half-integral spin, and bosons have integral spin) in the classic book of Streater & Wightman uses some results from SVC (Edge of the Wedge theorem and some analytic properties of Fourier transforms).